Physics:Chandrasekhar's white dwarf equation
In astrophysics, Chandrasekhar's white dwarf equation is an initial value ordinary differential equation introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar,[1] in his study of the gravitational potential of completely degenerate white dwarf stars. The equation reads as[2]
- [math]\displaystyle{ \frac 1 {\eta^2} \frac d {d\eta}\left(\eta^2 \frac{d\varphi}{d\eta}\right) + (\varphi^2 - C)^{3/2} = 0 }[/math]
with initial conditions
- [math]\displaystyle{ \varphi(0)=1, \quad \varphi'(0)=0 }[/math]
where [math]\displaystyle{ \varphi }[/math] measures the density of white dwarf, [math]\displaystyle{ \eta }[/math] is the non-dimensional radial distance from the center and [math]\displaystyle{ C }[/math] is a constant which is related to the density of the white dwarf at the center. The boundary [math]\displaystyle{ \eta=\eta_\infty }[/math] of the equation is defined by the condition
- [math]\displaystyle{ \varphi(\eta_\infty) = \sqrt{C}. }[/math]
such that the range of [math]\displaystyle{ \varphi }[/math] becomes [math]\displaystyle{ \sqrt{C}\leq\varphi\leq 1 }[/math]. This condition is equivalent to saying that the density vanishes at [math]\displaystyle{ \eta=\eta_\infty }[/math].
Derivation
From the quantum statistics of a completely degenerate electron gas (all the lowest quantum states are occupied), the pressure and the density of a white dwarf are calculated in terms of the maximum electron momentum [math]\displaystyle{ p_0 }[/math]standardized as [math]\displaystyle{ x = p_0 / mc }[/math], with pressure [math]\displaystyle{ P = A f(x) }[/math] and density [math]\displaystyle{ \rho = B x^3 }[/math], where
- [math]\displaystyle{ \begin{align} & A = \frac{\pi m_e^4 c^5}{3h^3} = 6.02\times 10^{21} \text{ Pa}, \\ & B = \frac{8\pi}{3}m_p\mu_e\left(\frac{m_e c}{h}\right)^3 = 9.82\times 10^8 \mu_e \text{ kg/m}^3, \\ & f(x) = x(2x^2-3)(x^2+1)^{1/2} + 3 \sinh^{-1} x, \end{align} }[/math]
[math]\displaystyle{ \mu_e }[/math] is the mean molecular weight of the gas, and [math]\displaystyle{ h }[/math] is the height of a small cube of gas with only two possible states.
When this is substituted into the hydrostatic equilibrium equation
- [math]\displaystyle{ \frac 1 {r^2} \frac{d}{dr}\left(\frac{r^2}{\rho}\frac{dP}{dr}\right)=-4\pi G \rho }[/math]
where [math]\displaystyle{ G }[/math] is the gravitational constant and [math]\displaystyle{ r }[/math] is the radial distance, we get
- [math]\displaystyle{ \frac{1}{r^2} \frac{d}{dr}\left(r^2\frac{d\sqrt{x^2+1}}{dr}\right)=-\frac{\pi G B^2}{2A}x^3 }[/math]
and letting [math]\displaystyle{ y^2 = x^2 + 1 }[/math], we have
- [math]\displaystyle{ \frac{1}{r^2} \frac{d}{dr}\left(r^2\frac{dy}{dr}\right)=-\frac{\pi G B^2}{2A}(y^2-1)^{3/2} }[/math]
If we denote the density at the origin as [math]\displaystyle{ \rho_o = B x_o^3 = B (y_o^2-1)^{3/2} }[/math], then a non-dimensional scale
- [math]\displaystyle{ r = \left(\frac{2A}{\pi G B^2}\right)^{1/2} \frac{\eta}{y_o}, \quad y = y_o \varphi }[/math]
gives
- [math]\displaystyle{ \frac{1}{\eta^2} \frac{d}{d\eta}\left(\eta^2 \frac{d\varphi}{d\eta}\right) + (\varphi^2 - C)^{3/2} = 0 }[/math]
where [math]\displaystyle{ C=1/y_o^2 }[/math]. In other words, once the above equation is solved the density is given by
- [math]\displaystyle{ \rho = B y_o^3 \left(\varphi^2 - \frac 1 {y_o^2}\right)^{3/2}. }[/math]
The mass interior to a specified point can then be calculated
- [math]\displaystyle{ M(\eta) = -\frac{4\pi}{B^2}\left(\frac{2A}{\pi G}\right)^{3/2}\eta^2\frac{d\varphi}{d\eta}. }[/math]
The radius-mass relation of the white dwarf is usually plotted in the plane [math]\displaystyle{ \eta_\infty }[/math]-[math]\displaystyle{ M(\eta_\infty) }[/math].
Solution near the origin
In the neighborhood of the origin, [math]\displaystyle{ \eta\ll 1 }[/math], Chandrasekhar provided an asymptotic expansion as
- [math]\displaystyle{ \begin{align} \varphi = {} & 1 - \frac{q^3} 6 \eta^2 + \frac{q^4}{40} \eta^4 - \frac{q^5(5q^2+14)}{7!} \eta^6 \\[6pt] & {} + \frac{q^6(339q^2 + 280)}{3\times 9!}\eta^8 - \frac{q^7(1425q^4 + 11346q^2 + 4256)}{5\times 11!}\eta^{10} + \cdots \end{align} }[/math]
where [math]\displaystyle{ q^2 = C-1 }[/math]. He also provided numerical solutions for the range [math]\displaystyle{ C = 0.01 - 0.8 }[/math].
Equation for small central densities
When the central density [math]\displaystyle{ \rho_o = B x_o^3 = B (y_o^2-1)^{3/2} }[/math] is small, the equation can be reduced to a Lane-Emden equation by introducing
- [math]\displaystyle{ \xi = \sqrt{2}\eta, \qquad \theta = \varphi^2-C= \varphi^2-1+x_o^2 + O(x_o^4) }[/math]
to obtain at leading order, the following equation
- [math]\displaystyle{ \frac{1}{\xi^2}\frac{d}{d\xi}\left(\xi^2\frac{d\theta}{d\xi}\right) = - \theta^{3/2} }[/math]
subjected to the conditions [math]\displaystyle{ \theta(0)=x_o^2 }[/math] and [math]\displaystyle{ \theta'(0)=0 }[/math]. Note that although the equation reduces to the Lane-Emden equation with polytropic index [math]\displaystyle{ 3/2 }[/math], the initial condition is not that of the Lane-Emden equation.
Limiting mass for large central densities
When the central density becomes large, i.e., [math]\displaystyle{ y_o\rightarrow \infty }[/math] or equivalently [math]\displaystyle{ C\rightarrow 0 }[/math], the governing equation reduces to
- [math]\displaystyle{ \frac{1}{\eta^2}\frac{d}{d\eta}\left(\eta^2\frac{d\varphi}{d\eta}\right) = - \varphi^{3} }[/math]
subjected to the conditions [math]\displaystyle{ \varphi(0)=1 }[/math] and [math]\displaystyle{ \varphi'(0)=0 }[/math]. This is exactly the Lane-Emden equation with polytropic index [math]\displaystyle{ 3 }[/math]. Note that in this limit of large densities, the radius
- [math]\displaystyle{ r = \left(\frac{2A}{\pi G B^2}\right)^{1/2} \frac{\eta}{y_o} }[/math]
tends to zero. The mass of the white dwarf however tends to a finite limit
- [math]\displaystyle{ M\rightarrow - \frac{4\pi}{B^2}\left(\frac{2A}{\pi G}\right)^{3/2}\left(\eta^2 \frac{d\varphi}{d\eta}\right)_{\eta=\eta_\infty}=5.75 \mu_e^{-2}M_\odot. }[/math]
The Chandrasekhar limit follows from this limit.
See also
References
- ↑ Chandrasekhar, Subrahmanyan, and Subrahmanyan Chandrasekhar. An introduction to the study of stellar structure. Vol. 2. Chapter 11 Courier Corporation, 1958.
- ↑ Davis, Harold Thayer (1962). Introduction to Nonlinear Differential and Integral Equations. Courier Corporation. ISBN 978-0-486-60971-3. https://books.google.com/books?id=RgbWowrjKd4C.
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